Anilox rollers are well known and German Patent Disclosure Document DE-OS 38 22 692 describes such a roller in which a roller or cylinder core carries a ceramic or metal layer of wear-resistant material. This layer is engraved to receive a layer ink accepting, water repelling or water rejecting, or hydrophobic material. Initially, the ink accepting layer is applied, with excess, on the roller which already carries the engraved layer of ceramic or metal wear-resistant material. The excess material of the ink accepting layer is then removed until the outer surfaces of the ribs of the wear-resistant material are free. The receptor depressions or cells are then placed or formed in the ink accepting layer between the freed ribs. Engraving the wear-resistant layer to form the cells leaving the ribs is carried out by means of a laser. This is a very expensive and time-consuming procedure.